Ergodic Estimations of Upscaled Coefficients for Diffusion in Random Velocity Fields

Upscaled coefficients for diffusion in ergodic velocity fields are derived by summing up correlations of increments of the position process, or equivalently of the Lagrangian velocity. Ergodic estimations of the correlations are obtained from time averages over finite paths sampled on a single trajectory of the process and a space average with respect to the initial positions of the paths. The first term in this path decomposition of the diffusion coefficients corresponds to Markovian diffusive behavior and is the only contribution for processes with independent increments. The next terms describe memory effects on diffusion coefficients until they level off to the value of the upscaled coefficients. Since the convergence with respect to the path length is rather fast and no repeated Monte Carlo simulations are required, this method speeds up the computation of the upscaled coefficients over methods based on long-time limit and ensemble averages by four orders of magnitude.